Exponential sums, quadratic splines and the Riemann zeta-function
The author presents a formula for the transformation of the exponential sum (\(a,b\) are integers with \(a<b\)) NEWLINE\[NEWLINES = \mathop{{\sum}'}\limits_{a\leq k \leq b}\gamma(k) \text{ e}^{2\pi if(k)}, NEWLINE\]NEWLINEwhere the dash \({'}\) means that the first and the last term in the above sum are to be halved, and the real valued functions \(\gamma \in \text{ C}^2[a,b],\;f\in\text{ C}^5[a,b]\) satisfy the following properties: there exists a constant \(c\geq 1\) and constants \(H>0, U\geq b-a\) and \(A\geq c^{-1}\) such that \(A \leq cU\) and NEWLINE\[NEWLINE\begin{aligned} & c^{-1}A^{-1} \le f''(x)\le cA^{-1},\quad|f^{(\ell)}(x)|\le cA^{-1}U^{2-\ell} \quad(\ell = 3,4,5),\\ &|\gamma^{(\ell)}(x)|\le cHU^{-\ell}\quad(\ell = 0,1,2, x\in[a, b]).NEWLINE\end{aligned} NEWLINE\]NEWLINEThese are the hypotheses of the kind needed for the application of the classical van der Corput process in the theory of the exponential sums \(S\). The novel idea of the author's approach is to approximate first \(\gamma\) and \(f\) by piecewise polynomial functions. This produces a result, whose proof is only sketched, which has an explicit expression for the error term, plus a new error term, which does not seem obtainable by classical methods. The formula in question, which is used to derive an approximate functional equation for \(\zeta(s)\) with a good error term, is NEWLINE\[NEWLINES=\text{e}^{i{\pi\over 4}}\sum_{[f'(a)]+1\leq k\leq[f'(b)]}{\gamma(x(k))\over{\sqrt{f''(x(k))}}} \text{e}^{2\pi i(f(x(k))-kx(k))}+R(b)-R(a)+O_c(H),\tag{1}NEWLINE\]NEWLINEwhere \(x(\cdot)\) is the unique function satisfying \(f'(x(y)) = y\) for \(y \in [f'(a), f'(b)]\), NEWLINE\[NEWLINER(\ell) = \gamma(\ell)\text{ e}^{2\pi i\ell}\phi(f''(\ell),\{f'(\ell)\}), \quad \{x\} = x - [x], NEWLINE\]NEWLINEand NEWLINE\[NEWLINE\phi(\lambda, \mu) := \int_0^\infty {\sinh(2\pi(\mu-{1\over 2})x)\over\sinh(\pi x)} \text{ e}^{i(\pi/2-\pi \lambda x^2)} \text{ d}x. NEWLINE\]NEWLINEThe main term (the sum on the right-hand side of (1)) is as in the classical van der Corput transformation, but the expressions for \(R(b) - R(a)\) are new.
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